Term 2 Week 5: Time Preferences and Sin Taxes

Question 1

What is the economic rationale for consumption taxes (3)

Answer 1

-Correct for harmful effects of consumption on third parties (pigovian taxes)
-Raise government revenue
-Reduce consumption of G/S with potential danger to health/wellbeing

Question 2

What is discounting and why does it occur (3,3)

Answer 2

-Decisions taken over time involve discounting
-We typically assume a constant discount rate δ
-if δ = 0.9, £100 next year is worth £90 etc.

Money in the future is discounted relative to today, as:
-Opportunity cost
-Patience
-Probability we actually get to period t

Question 3

What is present bias and how does hyperbolic discounting account for this (2,3)

Answer 3

-Present bias is when people tend to overvalue immediate rewards relative to all future rewards
-People have a preference to incur positive payoffs today, and postpone negative payoffs, only to reverse that preference when the future approaches

-Standard discounting measures total utility as U = u0 + δu1 + δ^2u2 …
-(Quasi)Hyperbolic discounting measures total utility as: U = u0 + δβu1 + δ^2βu2 …
-β < 1 indicates a present bias, β = 1 is no present bias

Question 4

What is the main government justification of sin taxes (1)

Answer 4

-One government justification of implementing a sin tax (alcohol, sugar etc) is that consumers have present bias, and will prioritise the benefits of consumption now over the health cost later

Question 5

How can we derive the implementation of sin taxes on the consumer maximisation problem (2,3,2)

Answer 5

-Consumers have discounted utility U = u0 + βδu1 + βδ^2u2 + …
-Here, β < 1 implies a present bias

-Say there are 3 ways for consumers to spend their budget (B), x = crisps, y = carrots, z = money
-At each time period the consumer solves maximisation problem: ln(x) + ln(y) + z - βδγln(x) s.t. B = (px+tx)x + (py + ty)y + z
-γ<1 measures the harm done in the next period of consuming crisps

The consumer maximisation problem will give demand for x:
-x* = (1-βδγ)/(px + tx)
-y* = 1/(py + ty)

Question 6

How can we derive the implementation of sin taxes with government action (1,3,2,3)

Answer 6

-First of all, normalise p_x = p_y = 1

-As a benevolent government, we maximise over t_x, t_y
-Ū = ln(x) + ln(y) + z* - δγln(x) s.t. t_xx + t_yy* = R
-Since the government is not present biased, they don’t have the β factor

Using the lagrangian, -1 + t_x* = μ(1-βδγ)/(1-δγ)
-1 + t_y* = μ

-This tells us t_x - t_y > 0 when (1-βδγ)/(1-δγ) > 1
-This happens iff β<1
-Hence, a present bias necessitates a larger tax on the sin good

Question 7

What is the slope of the indifference curve of goods x and y (gov vs consumer), and what does this tell us about their optimal consumption (2,1)

Answer 7

-The government’s IC slope = -MRS = -(1-δγ)y/x
-The government’s IC slope = -MRS = -(1-βδγ)y/x

-When plotting these both against a budget constraint, the governments optimal consumption point is less x and more y than the consumer

Question 8

What is the difference between sophisticates and Naifs (2,2)

Answer 8

-If some consumers had present bias and some didn’t, there will be a tradeoff between β < 1 and hurting β = 1
-The effect is larger on β < 1 types, as demand is higher

-Sophisticates would vote for this tax, understanding they have a lack of control
-Naifs would believe they would be fully self controlled in the future

Question 9

What are the pros and cons of sin taxes (2,4)

Answer 9

+Can combat serious public health problems
+Generates revenue by taxing inelastic products

-Could be regressive (UK sugar tax annual CV is 40% higher for bottom half of income distribution)
-Could encourage substitutes consumption
-Could encourage the black market
-Paternalistic (is it the governments place to intervene)